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As I understand, the number of all infinite lentgh sequences which is consist of $\left\{1,2\right\}$ is uncountable. I want to learn, Does mean of uncountable infinite equal to infinite, which is known in calculus?

I mean for example,

Let, $N$ be a number of all infinite lentgh sequences which is consist of $\left\{1,2\right\}$.

Then, can we say ?

$$\lim_{n\to\infty} \frac{N}{2^n}=1 $$

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No, these are completely different concepts. $N$ is not a number. There is no such thing as the "number of all infinite length sequences of $\{1,2\}$. What you could talk about is the cardinality of the set of all infinite length sequences of $\{1,2\}$.

However, if you define $N$ to be a cardinality, then the expression $\frac{N}{2^n}$ is undefined, and thus the limit you are asking for does not exist.

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  • $\begingroup$ I mean for example, $n=5$. We have $2^5$ possible finite sequence. Then if $n\to\infty$? $\endgroup$
    – user548054
    Jul 1, 2019 at 7:16
  • $\begingroup$ @Elementary If $n=5$, then the expression $\frac{N}{2^5}$ is nonsensical. $\endgroup$
    – 5xum
    Jul 1, 2019 at 7:18
  • $\begingroup$ math.stackexchange.com/q/3286436/548054 $\endgroup$
    – user548054
    Jul 8, 2019 at 9:05